I just got these and I love them.
This stool is pretty expensive for a stool but is a really good design for playing pins. A small back to lean back on if you want, and the round wheel base is a fair bit bigger than the normal chair - makes it nice to prop your feet up on while you play. Would be easy to roll between a few pins.
You could try a saddle style stool..
A stool you just lean against..
Low price point, comfy, and its a pair of stools..
Racing seat pair of bar stools..
Very comfy gaming chair. This company also makes gaming recliners. Of course a recliner wouldn't be good for sitting at a pin but if you do VR pinball with something like the oculus, it would be a very nice chair for that haha.
Anyhow best of luck. And if you find yourself unsure you could always just ask them what they would like :)